Since \(\Phi \)  and \(\Phi ^\ast \)  satisfy the \(\Delta _2\) -condition for large numbers holds, it follows that \(C^\infty (\bar{\Omega })\)  is dense in \(W^{1,\Phi }(\Omega )\)  and the the trace operator \(\gamma : W^{1,\Phi }(\Omega )\rightarrow L^\Phi (\Gamma )\) , defined by \(\gamma u:=u|_{\Gamma }\)  for all \(u\in C^\infty (\bar{\Omega })\)  is continuous. By [1]}, if (REF ) holds and \( (\Phi _\ast )^{(N-1)/N}\succ \succ \Phi \) , then the trace operator is also compact, i.e., the embedding \(W^{1,\Phi }(\Omega ) \hookrightarrow L^\Phi (\Gamma )\)  is compact.
